__Category__: Differential Equations"Published in Vacaville, California, USA"

Find the particular solution for

in which y = 3 when x = 1.

__Solution__:

Consider the given equation above

Did you notice that the given equation cannot be solved by separation of variables? The algebraic functions are the combination of x and y in the group and there's no way that we can separate x and y.

This type of differential equation is a homogeneous function. Let's consider this procedure in solving the given equation as follows

Let

so that

Substitute the values of y and dy to the given equation, we have

The resulting equation can now be separated by separation of variables as follows

Integrate on both sides of the equation, we have

Take the inverse natural logarithm on both sides of the equation, we have

But

Hence, the above equation becomes

Substitute the value of x and y in order to get the value of C as follows

Therefore, the particular solution is

There's another way in getting the particular solution of the given equation above as follows

Integrate both sides of the equation, we have

Substitute the value of x and y in order to get the value of C as follows

Therefore, the particular solution is